Well, quite often peeople quote research done in the 1950's that showed infantry enegagements were at shorter ranges than expected, etc., etc., etc...
I think a big part of the switch was not so much a 30-06 to 5.56, but rather a middle ground between 30-06 and .45ACP.
A rifle was very useful for its engagement range and killing power, and a submachingun was very useful for its high ammunition capacity, compactness, and enourmous volume of fire. IMHO the assault rifle cartridge was an attempt to capitalize on the best points of these two weapon systems, in effect giving every man a weapon that was both a rifle and a submachinegun all in one package.
This also in effect gave the advantage of making every man in the squad an automatic rifleman, as opposed to one man armed with a BAR or M14A1 and gave a lot more flexiblity to the squad.
It is funny how things have come full circle- people are now pushing for heavier cartridges, and it is very rare for M16's to be used on full auto- people are trained to use rapid strings of semiauto fire.